 reserve R for finite Approximation_Space;
 reserve X,Y,Z for Subset of R;
 reserve kap for RIF of R;

theorem Prop6d1: :: Proposition 6 d1)
  (CMap kappa_1 R).(X,Y) + (CMap kappa_1 R).(Y,Z) >= (CMap kappa_1 R).(X,Z)
  proof
    per cases;
    suppose
F1:   X = {}; then
      X c= Y; then
F2:   (CMap kappa_1 R).(X,Y) = 0 by Prop6a;
      (CMap kappa_1 R).(X,Z) = 0 by Prop6a,F1,XBOOLE_1:2;
      hence thesis by F2,XXREAL_1:1;
    end;
    suppose
F1:   X <> {} & Y = {} & Z <> {}; then
      Y c= Z; then
F2:   (CMap kappa_1 R).(Y,Z) = 0 by Prop6a;
      (CMap kappa_1 R).(X,Y) = 1 by F1,Ble1;
      hence thesis by F2,XXREAL_1:1;
    end;
    suppose
F1:   X <> {} & Y <> {} & Z = {}; then
F3:   (CMap kappa_1 R).(X,Z) = 1 by Ble1;
      (CMap kappa_1 R).(X,Y) >= 0 by XXREAL_1:1; then
      (CMap kappa_1 R).(X,Y) + 1 >= 0 + 1 by XREAL_1:6;
      hence thesis by F1,F3,Ble1;
    end;
    suppose
F1:   X <> {} & Y = {} & Z = {}; then
F3:   (CMap kappa_1 R).(X,Y) = 1 by Ble1;
      (CMap kappa_1 R).(Y,Z) >= 0 by XXREAL_1:1; then
      1 + (CMap kappa_1 R).(Y,Z) >= 1 + 0 by XREAL_1:6;
      hence thesis by F1,F3;
    end;
    suppose
      X <> {} & Y <> {} & Z <> {};
      hence thesis by Jaga1;
    end;
  end;
